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Young Children's Understanding of the Successor Function

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Young Children's Understanding of the Successor Function Young Children’s Understanding of the Successor Function Jennifer A. Kaminski ([email protected]) Department of Mathematics and Statistics, Wright State University Dayton, OH 45435 USA Abstract for numbers larger than one. Then children become “two- knowers”, correctly responding to requests for one or two This study examined 4-year-old children’s understanding of the successor function, the concept that for every positive items, but incorrectly for larger numbers. Subsequently, integer there is a unique next integer. Children were tested in they become “three-knowers” and possibly later “four- the context of cardinal numbers and ordinal numbers. The knower”. After several months as a “subset-knower” (for results suggest that knowledge of the successor for cardinal collections less than five), a child typically reaches a stage numbers precedes that for ordinal numbers. In addition, for where they can respond correctly to a request for any both cardinal and ordinal numbers, children generally failed number of items (i.e. cardinal-principle knower). to demonstrate understanding that the successor of a given number is unique. Some researchers have suggested that children transition from subset knowers to cardinal-principle knowers through Keywords: Counting; Natural Numbers; Cardinal Number; a bootstrapping process (Carey, 2004). Children who have Ordinal Number; Successor Function. knowledge of cardinality for collections of one, two, and three items extend this knowledge to larger collections of Introduction items by the inductive inference “If the number N represents A large amount of research has examined children’s the cardinality of a set of N items, then the next number in understanding of natural numbers (i.e. positive integers) and the count list, N+1, represents the cardinality of a set of N counting (e.g. Baroody, 1987; Fuson, 1988; Gelman & +1 items. Sarnecka and Carey (2008) describe this notion Gallistel, 1978; Piaget, 1952; Schaeffer, Eggleston, & Scott, of a “next number” representing cardinality of a set one unit 1974; Wynn, 1990, 1992). One interesting phenomenon that larger than currently under consideration as knowledge of has been observed is that children aged two to three years the mathematical concept of successor. They suggest that may appear at first glance to know how to count; they may cardinal-principle knowers have implicit knowledge of the correctly recite the count list (“one, two, three, four, …”) successor function that enables them to make the induction and even point to items in a collection and tag them with a from knowledge of cardinalities of small sets (e.g. less than number (e.g. Briars & Siegler, 1984; Fuson, 1988). four) to cardinalities of larger sets. They support their However, when asked “how many” items are in a collection argument by demonstrating that cardinal-principle knowers, they just “counted”, these children often do not know and not subset-knowers, recognize (1) that adding one item (Schaeffer, Eggleston, & Scott, 1974; Wynn, 1990, 1992). and not subtracting one item to a set of cardinality five Moreover, many children who can correctly answer the produces a set of cardinality six and (2) that adding one item “how many” question are unable to produce a collection of to a set of cardinality four results in five while adding two objects of a specific cardinality (Le Corre & Carey, 2007; items results in six. Wynn, 1992). For example, an experimenter might ask a While it is clear from these results that children who child to “give three toys to a puppet” and the child would understand the cardinality principle, as evidence by accurate give an incorrect number. Failure on these basic tasks performance on “Give-N” tasks, perform better on other suggests that young children (typically those under four numerical tasks than children who don’t understand the years of age) do not understand fundamental aspects of cardinality principle, these results do not necessarily imply natural numbers, including the Cardinality Principle. (see that cardinal-principle knowers have knowledge of the Gelman and Gallistel, 1978 for principles of counting) This successor function. The interpretation of successor that principle states that when counting a collection of items, the Sarnecka and Carey (2008) employ is essentially a notion of number associated with the final item counted represents the “there is a next”. This interpretation is inconsistent with a cardinality of the collection of items (i.e. the number of formal mathematical definition of successor function, and items). more importantly it is insufficient to generate the natural Accurate performance on such “Give-N” tasks has been numbers and hence explain the acquisition of the natural taken as evidence that children have knowledge of the numbers. As Lance, Asmuth, and Bloomfield (2005) point cardinality principle (Le Corre & Carey, 2007; Wynn, out, the simple notion that there is a “next” number can 1992). This knowledge emerges in a series of stages. First describe sets other than the natural numbers. For example, children respond by giving an arbitrary or an idiosyncratic the set of integers with addition modulo 12 (i.e. equivalent number of items (e.g. always a handful or always one). By to telling time on a standard clock face) has a successor age 2 ½ to 3 years, children typically respond correctly function; the successor of one is two, the successor of two is when asked to give one item, but give an incorrect number 1045 three, etc. However, unlike the natural numbers, the Mathematically, uniqueness can be proven by successor of twelve is one. demonstrating that for two natural numbers, a and b, if the The problem with the Sarnecka and Carey interpretation successor of a = the successor of b, then a = b. Knowledge is that they omitted two necessary characteristics of the of a unique successor can be tested in an analogous way by successor function. First, that the number one is not the stating a number and asking the participant what number successor of any number (if we do not include zero in the immediately precedes it. Two testing contexts were created, set of natural numbers). Second, the successor of a number one cardinal and one ordinal. Participants were tested on is unique. For example, the successor of two is only three; their ability to state a next number when given a number and three is the successor only of two. Specifically, the (i.e. successor). These questions are denoted as +1 natural numbers can be defined by the Dedekind/ Peano’s questions in the Method section. Participants were also Axioms that state the following (Dedekind, 1901). tested on their ability to state a preceding number when 1. One is a natural number. given a number (i.e. uniqueness of the successor). These 2. Every natural number has a successor that is a natural questions are denoted as -1 questions in the Method section. number. The cardinal context involved collections of circular disks 3. One is not the successor of any natural number. that were placed under a cup. Participants needed to state 4. Two natural numbers for which the successors are how many disks were under the cup when a disk was added equal are themselves equal (i.e. the successor of any or removed. The ordinal context involved a simple board natural number is unique). game in which participants were asked the number 5. If a set, S, of natural numbers contains one and for associated with locations before or after the location of a every k in S, the successor of k is also in S, then every game pawn (see Figure 1). natural number is in S. These axioms provide necessary and sufficient criteria to Experiment generate the set of natural numbers and therefore knowledge of these axioms could explain the acquisition of natural Method numbers. Participants Participants were 35 preschool children (21 The interpretation of successor used in the Sarnecka and girls, 14 boys, M = 3.86 years, SD = 0.15) recruited from Carey (2008) study does not demonstrate uniqueness of the preschools and childcare centers in the Columbus, Ohio area. successor function. Specifically, demonstrating that a child Materials and Design Participants were given a series of recognizes five as the cardinality of a set that contained four different tasks, How-Many task, Give-N task, Cup task, and and then another item was added, does not imply that the Game task. child knows that cardinality of five cannot be achieved by How-Many task. This task consisted of six questions and adding one to another set size. In our everyday lives, as involved black circular chips approximately 1 inch in adults we take this implication for granted, but we cannot diameter. For each question, the experimenter placed a assume that children do. Therefore demonstrating that collection of chips on a plastic plate approximately 7 inches children have knowledge of the successor without in diameter. The experimenter then showed the child the demonstrating its uniqueness cannot fully explain how collection of chips and asked, “Can you tell me how many children acquire knowledge of natural numbers. chips are here?” The questions presented collections of size Additionally, the concept of successor is an ordinal 2, 3, 4, 5, 6, and 8 in a pseudo-random order. concept, yet is has been tested in the context of cardinality. Give-N task. This task consisted of six questions testing the The goal of the present study was to examine the numbers 2, 3, 4, 5, 6, and 8 in a pseudo-random order. The conditions under which young children demonstrate experimenter placed a collection of ten chips in front of the knowledge of the successor function. If children who child. The experimenter then gave the child the plastic plate understand the cardinality principle demonstrate knowledge and asked, “Can you give me N chips on this plate?” of a unique successor for cardinal numbers, then perhaps, as Cup task (Cardinal Test).
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